Fast Stochastic Composite Minimization and an Accelerated Frank-Wolfe Algorithm under Parallelization

Code

• github.com/bpauld/pfw
  OfficialIn papernone★ 1

Abstract

We consider the problem of minimizing the sum of two convex functions. One of those functions has Lipschitz-continuous gradients, and can be accessed via stochastic oracles, whereas the other is "simple". We provide a Bregman-type algorithm with accelerated convergence in function values to a ball containing the minimum. The radius of this ball depends on problem-dependent constants, including the variance of the stochastic oracle. We further show that this algorithmic setup naturally leads to a variant of Frank-Wolfe achieving acceleration under parallelization. More precisely, when minimizing a smooth convex function on a bounded domain, we show that one can achieve an primal-dual gap (in expectation) in O(1/ ) iterations, by only accessing gradients of the original function and a linear maximization oracle with O(1/) computing units in parallel. We illustrate this fast convergence on synthetic numerical experiments.

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